A000182 -id:A000182 - cp's OEIS frontend

A258901 E.g.f. satisfies: A(x) = Integral 1 + A(x)^4 dx.

1, 24, 32256, 285272064, 8967114326016, 735868743566229504, 130778914961055994085376, 44390350317502907443360825344, 26290393222157669992962395876622336, 25377887922329300948014930852183837507584, 37855568618678541873143615775486954119570128896

Offset: 0

Paul D. Hanna, Jun 14 2015

nonn

			E.g.f.: A(x) = x + 24*x^5/5! + 32256*x^9/9! + 285272064*x^13/13! + 8967114326016*x^17/17! + 735868743566229504*x^21/21! +...
where Series_Reversion(A(x)) = x - x^5/5 + x^9/9 - x^13/13 + x^17/17 +...
Also, A(x) = S(x)/C(x) where
S(x) = x - 6*x^5/5! - 1764*x^9/9! - 7700616*x^13/13! - 147910405104*x^17/17! - 8310698364852576*x^21/21! +...+ A258900(n)*x^(4*n+1)/(4*n+1)! +...
C(x) = 1 - 6*x^4/4! - 1764*x^8/8! - 7700616*x^12/12! - 147910405104*x^16/16! - 8310698364852576*x^20/20! +...+ A258900(n)*x^(4*n)/(4*n)! +...
such that C(x)^4 + S(x)^4 = 1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..112
Guo-Niu Han, Jing-Yi Liu, Divisibility properties of the tangent numbers and its generalizations, arXiv:1707.08882 [math.CO], 2017. See Table for k = 4 p. 8.

Cf. A000182, A000831, A258900, A258880, A258925, A258927, A259112, A259113, A258970.

nmax=20; Table[CoefficientList[InverseSeries[Series[Integrate[1/(1+x^4),x],{x,0,4*nmax+1}],x],x][[4*n-2]] * (4*n-3)!, {n,1,nmax+1}] (* Vaclav Kotesovec, Jun 18 2015 *)

/* E.g.f. Series_Reversion( Integral 1/(1+x^4) dx ): */
{a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^4 + O(x^(4*n+2))) ) ); (4*n+1)!*polcoeff(A,4*n+1)}
for(n=0,20,print1(a(n),", "))

/* E.g.f. A(x) = Integral 1 + A(x)^4 dx.: */
{a(n) = local(A=x); for(i=1,n+1, A = intformal( 1 + A^4 + O(x^(4*n+2)) )); (4*n+1)!*polcoeff(A,4*n+1)}
for(n=0,20,print1(a(n),", "))

E.g.f. A(x) satisfies:
(1) A(x) = Series_Reversion( Integral 1/(1+x^4) dx ).
(2) A(x) = sqrt( tan( 2 * Integral A(x) dx ) ).
Let C(x) = S'(x) such that S(x) = Series_Reversion( Integral 1/(1-x^4)^(1/4) dx ) is the e.g.f. of A258900, then e.g.f. A(x) of this sequence satisfies:
(3) A(x) = S(x)/C(x),
(4) A(x) = Integral 1/C(x)^4 dx,
(5) A(x)^2 = S(x)^2/C(x)^2 = tan( 2 * Integral S(x)/C(x) dx ).
a(n) ~ 2^(6*n + 14/3) * (4*n)! * n^(1/3) / (3^(1/3) * Gamma(1/3) * Pi^(4*n + 4/3)). - Vaclav Kotesovec, Jun 18 2015

A296839 Expansion of e.g.f. tan(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, 33, 437, 22205, 978873, 81005113, 7356832669, 949918117653, 142805534055905, 27120922891214801, 6016195462632487941, 1592800634594574194413, 486576430503128985793417, 171866951067212728072402665, 69025662074064538734826793453

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			tan(x*tan(x/2)) = x^2/2! + x^4/4! + 33*x^6/6! + 437*x^8/8! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A000182, A001469, A003718, A009707, A024265, A110501, A296835, A296841, A296842.

nmax = 16; Table[(CoefficientList[Series[Tan[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] tan(x*tan(x/2)).

a(n) ~ c * d^n * n^(2*n + 1/2) / exp(2*n), where d = 16/Pi^2 = 1.621138938277404343102071411355642222469740394755... is the root of the equation tan(1/sqrt(d)) = Pi*sqrt(d)/4 and c = 1.75568815831... - Vaclav Kotesovec, Dec 21 2017, updated Mar 16 2024

A024283 E.g.f. (1/2) * tan(x)^2 (even powers only).

Original entry on oeis.org

0, 1, 8, 136, 3968, 176896, 11184128, 951878656, 104932671488, 14544442556416, 2475749026562048, 507711943253426176, 123460740095103991808, 35125800801971979943936, 11559592093904798920736768, 4356981378562584648085405696, 1864703851860264785548754812928

Offset: 0

N. J. A. Sloane. This sequence was in the 1973 "Handbook", but was then omitted from the database. Resubmitted by R. H. Hardin. Entry revised by N. J. A. Sloane, Jun 12 2012

Number of cyclically reverse alternating permutations of length 2n+2, cf. A024255. - Vladeta Jovovic, May 20 2007 [Comment corrected by Fausto A. C. Cariboni, Sep 02 2020]

Related to A102573: letting T(q,r) be the coefficient of n^r in the polynomial 2^(q-n)/n times sum(k=0..n binomial(n, k)*k^q), then A024283(x) = sum(k=0..(2*x-1) T(2*x,k)*(-1)^(k+x)*2^k). See Mathematica code below. [John M. Campbell, Sep 15 2013]

			(tan x)^2 = x^2 + 2/3*x^4 + 17/45*x^6 + 62/315*x^8 + ...
G.f. = x + 8*x^2 + 136*x^3 + 3968*x^4 + 176896*x^5 + 11184128*x^6 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259, T(n,2).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..100
Beáta Bényi, Miguel Méndez, José L. Ramírez and Tanay Wakhare, Restricted r-Stirling Numbers and their Combinatorial Applications, arXiv:1811.12897 [math.CO], 2018.

Cf. A009764, A024255.

Cf. A000182, A102573. A diagonal of A059419.

A024283 := n -> `if`(n=0,0,(-1)^(n-1)*2^(2*n+1)*polylog(-2*n-1,-1)); # Peter Luschny, Jun 28 2012

f[n_] := -(-1)^n 2^(2 n + 1) PolyLog[-1 - 2 n, -1]; f[0] = 0; Array[f, 15, 0] (* Robert G. Wilson v, Jun 28 2012 *)
poly[q_] := 2^(q-n)/n*FunctionExpand[Sum[Binomial[n, k]*k^q, {k, 0, n}]]; T[q_, r_] := First[Take[CoefficientList[poly[q], n], {r+1, r+1}]]; Print[Table[Sum[T[2*x, k]*(-1)^(k+ x)*(2^k), {k, 0, 2*x-1}], {x, 1, 10}]]; (* John M. Campbell, Sep 15 2013 *)
a[ n_] := If[ n < 1, 0, With[ {k = 2 n + 1}, k! SeriesCoefficient[ Tan[x] / 2, {x, 0, k}]]] (* Michael Somos, Jan 21 2014 *)
a[ n_] := If[ n < 0, 0, With[ {k = 2 n}, k! SeriesCoefficient[ Tan[x]^2 / 2, {x, 0, k}]]] (* Michael Somos, Jan 21 2014 *)
a[0] = 0; a[n_] := (4^(n+1)-1)*Gamma[2*(n+1)]*Zeta[2*(n+1)]/Pi^(2*(n+1)); Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 05 2016 *)

{a(n)=polcoeff( sum(m=1, n, x^m*prod(k=1, m, (2*k-1)^2/(1+(2*k-1)^2*x +x*O(x^n))) ), n)} \\ Paul D. Hanna, Feb 01 2013

G.f.: (1/2)*(tan(z))^2 = (z^2/(1-z^2)/2)*(1 +2*z^2/((z^2-1)*(G(0)-2*z^2)), G(k) = (k+2)*(2*k+3)-2*z^2+2*z^2*(k+2)*(2*k+3)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011
a(n) = (-1)^(n-1)*2^(2*n+1)*PolyLog(-2*n-1,-1) for n >= 1. - Peter Luschny, Jun 28 2012
O.g.f.: Sum_{n>=1} x^n * Product_{k=1..n} (2*k-1)^2 / (1 + (2*k-1)^2*x). - Paul D. Hanna, Feb 01 2013
G.f.: x/(Q(0)-x), where Q(k) = 1 + 2*x*(2*k+1)^2 - x*(2*k+3)^2*(1+x*(2*k+1)^2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2013
a(n) ~ (2*n)! * n * 2^(2*n+3) / Pi^(2*n+2). - Vaclav Kotesovec, Aug 22 2014
a(n) = (4^(n+1)-1)*Gamma(2*(n+1))*zeta(2*(n+1))/Pi^(2*(n+1)) for n >= 1. - Jean-François Alcover, Feb 05 2016
From Peter Bala, Nov 16 2020: (Start)
a(n) = (1/2)*A000182(n+1) for n >= 1.
Conjectural o.g.f.: x/(1 + x - 9*x/(1 - 8*x/(1 + x - 25*x/(1 - 24*x/(1 + x - ... - (2*n+1)^2*x/(1 - 4*n*(n+1)*x/(1 + x - ... ))))))). (End)
a(n) = (-1)^(n-1)*PolyLog(-2*n - 1, i) for n >= 1. - Peter Luschny, Aug 12 2021

Extended and signs tested Mar 15 1997.

A117972 Numerator of zeta'(-2n), n >= 0.

Original entry on oeis.org

1, -1, 3, -45, 315, -14175, 467775, -42567525, 638512875, -97692469875, 9280784638125, -2143861251406875, 147926426347074375, -48076088562799171875, 9086380738369043484375, -3952575621190533915703125

Offset: 0

Eric W. Weisstein, Apr 06 2006

In A160464 the coefficients of the ES1 matrix are defined. This matrix led to the discovery that the successive differences of the ES1[1-2*m,n] coefficients for m = 1, 2, 3, ..., are equal to the values of zeta'(-2n), see also A094665 and A160468. - Johannes W. Meijer, May 24 2009
A048896(n), n >= 1: Numerators of Maclaurin series for 1 - ((sin x)/x)^2,
  a(n), n >= 2: Denominators of Maclaurin series for 1 - ((sin x)/x)^2, the correlation function in Montgomery's pair correlation conjecture. - Daniel Forgues, Oct 16 2011
From Andrey Zabolotskiy, Sep 23 2021: (Start)
zeta'(-2n), which is mentioned in the Name, is irrational. For n > 0, a(n) is the numerator of the rational fraction g(n) = Pi^(2n)*zeta'(-2n)/zeta(2n+1). The denominator is 4*A048896(n-1). g(n) = f(n) for n > 0, where f(n) is given in the Formula section. Also, f(n) = Bernoulli(2n)/z(n)/4 (see Formula section) for all n.
For n = 0, zeta'(0) = -log(2Pi)/2, g(0) can be set to 0 because of the infinite denominator. However, a(0) is set to 1 because it is the numerator of f(0).
It seems that -4*f(n)*alpha_n = A000182(n), where alpha_n = A191657(n, p(n)) / A191658(n, p(n)) [where p(n) = A000041(n)] is the n-th "elementary coefficient" from the paper by Izaurieta et al. (End)

			-1/4, 3/4, -45/8, 315/4, -14175/8, 467775/8, -42567525/16, ...
-zeta(3)/(4*Pi^2), (3*zeta(5))/(4*Pi^4), (-45*zeta(7))/(8*Pi^6), (315*zeta(9))/(4*Pi^8), (-14175*zeta(11))/(8*Pi^10), ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Fernando Izaurieta, Ricardo Ramírez and Eduardo Rodríguez, Dirac Matrices for Chern-Simons Gravity, arXiv:1106.1648 [math-ph], 2011-2012.
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function

Cf. A000367, A002445, A049606, A046988, A002432, A117973, A048896.
From Johannes W. Meijer, May 24 2009: (Start)
Cf. A160464, A094665 and A160468.
Absolute values equal row sums of A160468. (End)
Cf. A191657, A191658, A000182, A000041.

# Without rational arithmetic
a := n -> (-1)^n*(2*n)!*2^(add(i,i=convert(n,base,2))-2*n);
# Peter Luschny, May 02 2009

Table[Numerator[(2 n)!/2^(2 n + 1) (-1)^n], {n, 0, 30}]

L:taylor(1/x*sin(sqrt(x))^2,x,0,15); makelist(denom(coeff(L,x,n))*(-1)^(n+1),n,0,15); /* Vladimir Kruchinin, May 30 2011 */

a(n) = numerator(f(n)) where f(n) = (2*n)!/2^(2*n + 1)(-1)^n, from the Mathematica code.
From Terry D. Grant, May 28 2017: (Start)
|a(n)| = A049606(2n).
a(n) = -numerator(Bernoulli(2n)/z(n)) where Bernoulli(2n) = A000367(n) / A002445(n) and z(n) = A046988(n) / A002432(n) for n > 0. (End) [Corrected by Andrey Zabolotskiy, Sep 23 2021]

First term added, offset changed and edited by Johannes W. Meijer, May 15 2009

A144696 Triangle of 2-Eulerian numbers.

Original entry on oeis.org

1, 1, 2, 1, 7, 4, 1, 18, 33, 8, 1, 41, 171, 131, 16, 1, 88, 718, 1208, 473, 32, 1, 183, 2682, 8422, 7197, 1611, 64, 1, 374, 9327, 49780, 78095, 38454, 5281, 128, 1, 757, 30973, 264409, 689155, 621199, 190783, 16867, 256

Offset: 2

Peter Bala, Sep 19 2008

Let [n] denote the ordered set {1,2,...,n}. The symmetric group S_n consists of the injective mappings p:[n] -> [n]. Such a permutation p has an excedance at position i, 1 <= i < n, if p(i) > i. One well-known interpretation of the Eulerian numbers A(n,k) is that they count the permutations in the symmetric group S_n with k excedances. The triangle of Eulerian numbers is A008292 (but with an offset of 1 in the column numbering). We generalize this definition to restricted permutations as follows.
Let r be a nonnegative integer and let Permute(n,n-r) denote the set of injective maps p:[n-r] -> [n], which we think of as permutations of n numbers taken n-r at a time. Clearly, |Permute(n,n-r)| = n!/r!. We say that p has an excedance at position i, 1 <= i <= n-r, if p(i) > i. Then the r-Eulerian number, denoted by A(r;n,k), is defined as the number of permutations in Permute(n,n-r) having k excedances. Thus the current array of 2-Eulerian numbers gives the number of permutations in Permute(n,n-2) with k excedances. See the example section below for some numerical examples.
Clearly A(0;n,k) = A(n,k). The case r = 1 also produces the ordinary Eulerian numbers A(n,k). There is an obvious bijection from Permute(n,n) to Permute(n,n-1) that preserves the number of excedances of a permutation. Consequently, the 1-Eulerian numbers are equal to the 0-Eulerian numbers: A(1;n,k) = A(0;n,k) = A(n,k).
For other cases of r-Eulerian numbers see A144697 (r = 3), A144698 (r = 4) and A144699 (r = 5). There is also a concept of r-Stirling numbers of the first and second kinds - see A143491 and A143494. If we multiply the entries of the current array by a factor of 2 and then reverse the rows we obtain A120434.
An alternative interpretation of the current array due to [Strosser] involves the 2-excedance statistic of a permutation (see also [Foata & Schutzenberger, Chapitre 4, Section 3]). We define a permutation p in Permute(n,n-2) to have a 2-excedance at position i (1 <=i <= n-2) if p(i) >= i + 2.
Given a permutation p in Permute(n,n-2), define ~p to be the permutation in Permute(n,n-2) that takes i to n+1 - p(n-i-1). The map ~ is a bijection of Permute(n,n-2) with the property that if p has (resp. does not have) an excedance in position i then ~p does not have (resp. has) a 2-excedance at position n-i-1. Hence ~ gives a bijection between the set of permutations with k excedances and the set of permutations with (n-k) 2-excedances. Thus reading the rows of this array in reverse order gives a triangle whose entries count the permutations in Permute(n,n-2) with k 2-excedances.
Example: Represent a permutation p:[n-2] -> [n] in Permute(n,n-2) by its image vector (p(1),...,p(n-2)). In Permute(10,8) the permutation p = (1,2,4,7,10,6,5,8) does not have an excedance in the first two positions (i = 1 and 2) or in the final three positions (i = 6, 7 and 8). The permutation ~p = (3,6,5,1,4,7,9,10) has 2-excedances only in the first three positions and the final two positions.
From Peter Bala, Dec 27 2019: (Start)
This is the array A(1,1,3) in the notation of Hwang et al. (p. 25), where the authors remark that the r-Eulerian numbers were first studied by Shanlan Li (Duoji Bilei, Ch. 4), who gave the summation formulas
Sum_{i = 2..n+1} (i-1)*C(i,2) = C(n+3,4) + 2*C(n+2,4)
Sum_{i = 2..n+1} (i-1)^2*C(i,2) = C(n+4,5) + 7*C(n+3,5) + 4*C(n+2,5)
Sum_{i = 2..n+1} (i-1)^3*C(i,2) = C(n+5,6) + 18*C(n+4,6) + 33*C(n+3,6) + 8*C(n+2,6). (End)

			The triangle begins
===========================================
n\k|..0.....1.....2.....3.....4.....5.....6
===========================================
2..|..1
3..|..1.....2
4..|..1.....7.....4
5..|..1....18....33.....8
6..|..1....41...171...131....16
7..|..1....88...718..1208...473....32
8..|..1...183..2682..8422..7197..1611....64
...
Row 4 = [1,7,4]: We represent a permutation p:[n-2] -> [n] in Permute(n,n-2) by its image vector (p(1),...,p(n-2)). Here n = 4. The permutation (1,2) has no excedances; 7 permutations have a single excedance, namely, (1,3), (1,4), (2,1), (3,1), (3,2), (4,1) and (4,2); the remaining 4 permutations, (2,3), (2,4), (3,4) and (4,3) each have two excedances.

J. Riordan. An introduction to combinatorial analysis. New York, J. Wiley, 1958.
R. Strosser. Séminaire de théorie combinatoire, I.R.M.A., Université de Strasbourg, 1969-1970.
Li, Shanlan (1867). Duoji bilei (Series summation by analogies), 4 scrolls. In Zeguxizhai suanxue (Mathematics from the Studio Devoted to the Imitation of the Ancient Chinese Tradition) (Jinling ed.), Volume 4.
Li, Shanlan (2019). Catégories analogues d’accumulations discrètes (Duoji bilei), traduit et commenté par Andrea Bréard. La Bibliothèque Chinoise. Paris: Les Belles Lettres.

G. C. Greubel, Rows n = 2..52 of the triangle, flattened
J. F. Barbero G., J. Salas, and E. J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv preprint arXiv:1307.5624 [math.CO], 2013-2015.
Mark Conger, A refinement of the Eulerian polynomials and the joint distribution of pi(1) and Des(pi) in S_n, arXiv:math/0508112 [math.CO], 2005.
Ming-Jian Ding and Bao-Xuan Zhu, Some results related to Hurwitz stability of combinatorial polynomials, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 9.
Sergi Elizalde, Descents on quasi-Stirling permutations, arXiv:2002.00985 [math.CO], 2020.
D. Foata and M. Schutzenberger, Théorie Géométrique des Polynômes Eulériens, Lecture Notes in Math., no.138, Springer Verlag 1970; arXiv:math/0508232 [math.CO], 2005.
Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018-2019.
Tanya Khovanova and Rich Wang, Ending States of a Special Variant of the Chip-Firing Algorithm, arXiv:2302.11067 [math.CO], 2023.
L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
Carla D. Savage and Gopal Viswanathan, The 1/k-Eulerian polynomials, Elec. J. of Comb., Vol. 19, Issue 1, #P9 (2012). - From _N. J. A. Sloane_, Feb 06 2013

Cf. A008292, A120434, A143491, A143494, A143497, A144697, A144698, A144699.
Cf. A000079 (right diagonal), A001710 (row sums).
Cf. A000182 (related to alt. row sums).

m:=2; [(&+[(-1)^(k-j)*Binomial(n+1,k-j)*Binomial(j+m,m-1)*(j+1)^(n-m+1): j in [0..k]])/m: k in [0..n-m], n in [m..m+10]]; // G. C. Greubel, Jun 04 2022

with(combinat):
T:= (n,k) -> 1/2!*add((-1)^(k-j)*binomial(n+1,k-j)*(j+1)^(n-1)*(j+2), j = 0..k):
for n from 2 to 10 do
seq(T(n,k),k = 0..n-2)
end do;

T[n_, k_]:= 1/2!*Sum[(-1)^(k-j)*Binomial[n+1, k-j]*(j+1)^(n-1)*(j+2), {j, 0, k}];
Table[T[n, k], {n,2,10}, {k,0,n-2}]//Flatten (* Jean-François Alcover, Oct 15 2019 *)

m=2 # A144696
def T(n,k): return (1/m)*sum( (-1)^(k-j)*binomial(n+1,k-j)*binomial(j+m,m-1)*(j+1)^(n-m+1) for j in (0..k) )
flatten([[T(n,k) for k in (0..n-m)] for n in (m..m+10)]) # G. C. Greubel, Jun 04 2022

T(n,k) = (1/2!)*Sum_{j = 0..k} (-1)^(k-j)*binomial(n+1,k-j)*(j+1)^(n-1)*(j+2);
T(n,n-k) = (1/2!)*Sum_{j = 2..k} (-1)^(k-j)*binomial(n+1,k-j)*j^(n-1)*(j-1).
Recurrence relations:
T(n,k) = (k+1)*T(n-1,k) + (n-k)*T(n-1,k-1) with boundary conditions T(n,0) = 1 for n >= 2, T(2,k) = 0 for k >= 1. Special cases: T(n,n-2) = 2^(n-2); T(n,n-3) = A066810(n-1).
E.g.f. (with suitable offsets): (1/2)*[(1 - x)/(1 - x*exp(t - t*x))]^2 = 1/2 + x*t + (x + 2*x^2)*t^2/2! + (x + 7*x^2 + 4*x^3)*t^3/3! + ... .
The row generating polynomials R_n(x) satisfy the recurrence R_(n+1)(x) = (n*x+1)*R_n(x) + x*(1-x)*d/dx(R_n(x)) with R_2(x) = 1. It follows that the polynomials R_n(x) for n >= 3 have only real zeros (apply Corollary 1.2. of [Liu and Wang]).
The (n+1)-th row generating polynomial = (1/2!)*Sum_{k = 1..n} (k+1)!*Stirling2(n,k) *x^(k-1)*(1-x)^(n-k).
For n >= 2,
(1/2)*(x*d/dx)^(n-1) (1/(1-x)^2) = x/(1-x)^(n+1) * Sum_{k = 0..n-2} T(n,k)*x^k,
(1/2)*(x*d/dx)^(n-1) (x^2/(1-x)^2) = 1/(1-x)^(n+1) * Sum_{k = 2..n} T(n,n-k)*x^k,
1/(1-x)^(n+1)*Sum_{k = 0..n-2} T(n,k)*x^k = (1/2!) * Sum_{m = 0..inf} (m+1)^(n-1)*(m+2)*x^m,
1/(1-x)^(n+1)*Sum_{k = 2..n} T(n,n-k)*x^k = (1/2!) * Sum_{m = 2..inf} m^(n-1)*(m-1)*x^m.
Worpitzky-type identities:
Sum_{k = 0..n-2} T(n,k)*binomial(x+k,n) = (1/2!)*x^(n-1)*(x - 1);
Sum_{k = 2..n} T(n,n-k)*binomial(x+k,n) = (1/2!)*(x + 1)^(n-1)*(x + 2).
Relation with Stirling numbers (Frobenius-type identities):
T(n+1,k-1) = (1/2!) * Sum_{j = 0..k} (-1)^(k-j)*(j+1)!* binomial(n-j,k-j)*Stirling2(n,j) for n,k >= 1;
T(n+1,k-1) = (1/2!) * Sum_{j = 0..n-k} (-1)^(n-k-j)*(j+1)!* binomial(n-j,k)*S(2;n+2,j+2) for n,k >= 1 and
T(n+2,k) = (1/2!) * Sum_{j = 0..n-k} (-1)^(n-k-j)*(j+2)!* binomial(n-j,k)*S(2;n+2,j+2) for n,k >= 0, where S(2;n,k) denotes the 2-Stirling numbers A143494(n,k).
The row polynomials of this array are related to the Eulerian polynomials. For example, 1/2*x*d/dx [x*(x + 4*x^2 + x^3)/(1-x)^4] = x^2*(1 + 7*x + 4*x^2)/(1-x)^5 and 1/2*x*d/dx [x*(x + 11*x^2 + 11*x^3 + x^4)/(1-x)^5] = x^2*(1 + 18*x + 33*x^2 + 8*x^3)/(1-x)^6.
Row sums A001710. Alternating row sums [1, -1, -2, 8, 16, -136, -272, 3968, 7936, ... ] are alternately (signed) tangent numbers and half tangent numbers - see A000182.
Sum_{k = 0..n-2} 2^k*T(n,k) = A069321(n-1). Sum_{k = 0..n-2} 2^(n-k)*T(n,k) = 4*A083410(n-1).
For n >=2, the shifted row polynomial t*R(n,t) = (1/2)*D^(n-1)(f(x,t)) evaluated at x = 0, where D is the operator (1-t)*(1+x)*d/dx and f(x,t) = (1+x*t/(t-1))^(-2). - Peter Bala, Apr 22 2012

A258927 E.g.f. satisfies: A(x) = Integral 1 + A(x)^6 dx.

Original entry on oeis.org

1, 720, 410572800, 4492717498368000, 348990783113936240640000, 118162808964225967251573964800000, 130226468530398571130647349959852032000000, 384446125794905598149974467971605129718661120000000, 2644398446216951886577241780697447635225293650237849600000000

Offset: 0

Paul D. Hanna, Jun 15 2015

nonn

From Vaclav Kotesovec, Jun 17 2015: (Start)
In general, for k>2, if e.g.f. satisfies A(x) = Integral 1 + A(x)^k dx, then a(n) ~ k^(k/(k-1)) * n^(1/(k-1)) * (k*n)! * (k*sin(Pi/k)/Pi)^(k*n + k/(k-1)) / ((k-1)^(1/(k-1)) * Gamma(1/(k-1))).
(End)

			E.g.f.: A(x) = x + 720*x^7/7! + 410572800*x^13/13! + 4492717498368000*x^19/19! +...
where Series_Reversion(A(x)) = x - x^7/7 + x^13/13 - x^19/19 + x^25/25 +...
Also, A(x) = S(x)/C(x) where
S(x) = x - 120*x^7/7! - 21859200*x^13/13! - 131273353728000*x^19/19! +...+ A258926(n)*x^(6*n+1)/(6*n+1)! +...
C(x) = 1 - 120*x^6/6! - 21859200*x^12/12! - 131273353728000*x^18/18! +...+ A258926(n)*x^(6*n)/(6*n)! +...
such that C(x)^6 + S(x)^6 = 1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..75

Cf. A000182(n-1) (k=2), A258880 (k=3), A258901 (k=4), A258925 (k=5), A259112 (k=7), A259113 (k=8), A258926, A258994.

/* E.g.f. Series_Reversion( Integral 1/(1+x^6) dx ): */
{a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^6 + O(x^(6*n+2))) ) ); (6*n+1)!*polcoeff(A, 6*n+1)}
for(n=0, 20, print1(a(n), ", "))

/* E.g.f. A(x) = Integral 1 + A(x)^6 dx.: */
{a(n) = local(A=x); for(i=1, n+1, A = intformal( 1 + A^6 + O(x^(6*n+2)) )); (6*n+1)!*polcoeff(A, 6*n+1)}
for(n=0, 20, print1(a(n), ", "))

E.g.f. A(x) satisfies:
(1) A(x) = Series_Reversion( Integral 1/(1+x^6) dx ).
(2) A(x)^3 = tan( 3 * Integral A(x)^2 dx ).
Let C(x) = S'(x) such that S(x) = Series_Reversion( Integral 1/(1-x^6)^(1/6) dx ) is the e.g.f. of A258926, then e.g.f. A(x) of this sequence satisfies:
(3) A(x) = S(x)/C(x),
(4) A(x) = Integral 1/C(x)^6 dx,
(5) A(x)^3 = S(x)^3/C(x)^3 = tan( 3 * Integral S(x)^2/C(x)^2 dx ).
a(n) ~ 2^(6/5) * 3^(6*n+12/5) * (6*n)! * n^(1/5) / (5^(1/5) * Gamma(1/5) * Pi^(6*n+6/5)). - Vaclav Kotesovec, Jun 18 2015

A293951 Number of linear extensions of a poset whose Hasse diagram consists of n binary shrubs with type B_n joins.

Original entry on oeis.org

1, 9, 477, 74601, 25740261, 16591655817, 17929265150637, 30098784753112329, 74180579084559895221, 256937013876000351610089, 1208025937371403268201735037, 7494692521096248546330688437801, 59931935202159196095445595508666501

Offset: 0

N. J. A. Sloane, Oct 29 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 0..100
Jeffrey Remmel, S. Zheng, Rises in forests of binary shrubs, arXiv preprint arXiv:1611.09018 [math.CO], 2016-2017.

Cf. A293950, A293952, A293953.

Cf. A000182 (m=2), this seq (m=3), A273352 (m=4), A318258 (m=5).

LMLlist[m_, len_] := Table[(-1)^(n + 1) (m n)!, {n, 1, len}]*
Delete [CoefficientList[Series[Log[MittagLefflerE[m, z]], {z, 0, len}], z], 1];
LMLlist[3, 13] (* Peter Luschny, Aug 26 2018 *)

Terms a(11) and beyond from Lars Blomberg, Oct 31 2017

A302587 a(n) = n! * [x^n] exp(nx)tanh(x).

Original entry on oeis.org

0, 1, 4, 25, 224, 2641, 38592, 671665, 13548544, 310580161, 7971353600, 226406902921, 7049219383296, 238722074157841, 8735529994928128, 343474252543881313, 14441163232204292096, 646510839624706118401, 30704150325602206089216, 1541807339347429264648441

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..386
N. J. A. Sloane, Transforms

Cf. A000182, A009832, A062024, A302583, A302584, A302585, A302586.

N:= 100: # to get a(0)..a(N)
T:= series(tanh(x),x,N+1):
C:= [seq(coeff(T,x,j),j=1..N)]:
seq(n! * add(C[i]*n^(n-i)/(n-i)!,i=1..n,2), n=0..N); # Robert Israel, Apr 10 2018

Table[n! SeriesCoefficient[Exp[n x] Tanh[x], {x, 0, n}], {n, 0, 19}]

a(n) = my(x='x+O('x^(n+1))); polcoeff(n!*exp(n*x)*tanh(x), n); \\ Michel Marcus, Apr 11 2018; corrected Jun 15 2022

a(n) ~ tanh(1) * n^n. - Vaclav Kotesovec, Jun 08 2019

A353611 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + tan(x).

Original entry on oeis.org

1, 0, 2, -8, 56, -336, 3184, -27264, 309760, -3297280, 48104704, -624745472, 10591523840, -159594803200, 3133776259072, -56224864108544, 1249919350046720, -24600643845095424, 624022403933077504, -14094091678163140608, 381632216575339397120, -9516741266133420605440

Offset: 1

Ilya Gutkovskiy, May 07 2022

sign

Cf. A000182, A006973, A009006, A137852, A353583, A353584, A353607, A353608, A353609, A353610.

nn = 22; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Tan[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A119468 Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(n,2j)*binomial(n-2j,k).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 8, 16, 12, 4, 1, 16, 40, 40, 20, 5, 1, 32, 96, 120, 80, 30, 6, 1, 64, 224, 336, 280, 140, 42, 7, 1, 128, 512, 896, 896, 560, 224, 56, 8, 1, 256, 1152, 2304, 2688, 2016, 1008, 336, 72, 9, 1, 512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 10, 1

Offset: 0

Paul Barry, May 21 2006

Product of Pascal's triangle A007318 and A119467. Row sums are A007051. Diagonal sums are A113225.
Variant of A080928, A115068 and A082137. - R. J. Mathar, Feb 09 2010
Matrix inverse of the Euler tangent triangle A081733. - Peter Luschny, Jul 18 2012
Central column: T(2*n,n) = A069723(n). - Peter Luschny, Jul 22 2012
Subtriangle of the triangle in A198792. - Philippe Deléham, Nov 10 2013

			Triangle begins
    1;
    1,    1;
    2,    2,    1;
    4,    6,    3,    1;
    8,   16,   12,    4,    1;
   16,   40,   40,   20,    5,    1;
   32,   96,  120,   80,   30,    6,    1;
   64,  224,  336,  280,  140,   42,    7,   1;
  128,  512,  896,  896,  560,  224,   56,   8,  1;
  256, 1152, 2304, 2688, 2016, 1008,  336,  72,  9,  1;
  512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 10, 1;

A082137 read as triangle with rows reversed.

Cf. A000182, A009006, A038207, A081733, A155585.

A119468_row := proc(n) local s,t,k;
  s := series(exp(z*x)/(1-tanh(x)),x,n+2);
  t := factorial(n)*coeff(s,x,n); seq(coeff(t,z,k), k=(0..n)) end:
for n from 0 to 7 do A119468_row(n) od; # Peter Luschny, Aug 01 2012
# Alternatively:
T := (n, k) -> 2^(n-k-1+0^(n-k))*binomial(n,k):
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, Nov 10 2017

A[k_] := Table[If[m < n, 1, -1], {m, k}, {n, k}]; a = Join[{{1}}, Table[(-1)^n*CoefficientList[CharacteristicPolynomial[A[n], x], x], {n, 1, 10}]]; Flatten[a] (* Roger L. Bagula and Gary W. Adamson, Jan 25 2009 *)
Table[Sum[Binomial[n,2j]Binomial[n-2j,k],{j,0,n-k}],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Dec 14 2022 *)

R = PolynomialRing(QQ, 'x')
def p(n,x) :
  return 1 if n==0 else add((-1)^n*binomial(n,k)*(x^(n-k)-1) for k in range(n))
def A119468_row(n):
    x = R.gen()
    return [abs(cf) for cf in list((p(n,x-1)-p(n,x+1))/2+x^n)]
for n in (0..8) : print(A119468_row(n)) # Peter Luschny, Jul 22 2012

G.f.: (1 - x - xy)/(1 - 2x - 2x*y + 2x^2*y + x^2*y^2).
Number triangle T(n,k) = Sum_{j=0..n} binomial(n,j)*binomial(j,k)*(1+(-1)^(j-k))/2.
Define matrix: A(n,m,k) = If[m < n, 1, -1];
  p(x,k) = CharacteristicPolynomial[A[n,m,k],x]; then t(n,m) = coefficients(p(x,n)). - Roger L. Bagula and Gary W. Adamson, Jan 25 2009
E.g.f.: exp(x*z)/(1-tanh(x)). - Peter Luschny, Aug 01 2012
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) - T(n-2,k-2) for n >= 2, T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 10 2013
E.g.f.: [(e^(2t)+1)/2] e^(tx) = e^(P.(x)t), so this is an Appell sequence with lowering operator D = d/dx and raising operator R = x + 2/(e^(-2D)+1), i.e., D P_n(x) = n P_{n-1}(x) and R p_n(x) = P_{n+1}(x) where P_n(x) = [(x+2)^n + x^n]/2. Also, (P.(x)+y)^n = P_n(x+y), umbrally. R = x + 1 + D - 2 D^3/3! + ... contains the e.g.f.(D) mod signs of A009006 and A155585 and signed, aerated A000182, the zag numbers, so the unsigned differential component 2/[e^(2D)+1] = 2 Sum_{n >= 0} Eta(-n) (-2D)^n/n!, where Eta(s) is the Dirichlet eta function, and 2 *(-2)^n Eta(-n) = (-1)^n (2^(n+1)-4^(n+1)) Zeta(-n) = (2^(n+1)-4^(n+1)) B(n+1)/(n+1) with Zeta(s), the Riemann zeta function, and B(n), the Bernoulli numbers. The polynomials PI_n(x) of A081733 are the umbral compositional inverses of this sequence, i.e., P_n(PI.(x)) = x^n = PI_n(P.(x)) under umbral composition. Aside from the signs and the main diagonals, multiplying this triangle by 2 gives the face-vectors of the hypercubes A038207. - Tom Copeland, Sep 27 2015
T(n,k) = 2^(n-k-1+0^(n-k))*binomial(n, k). - Peter Luschny, Nov 10 2017

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A258901 E.g.f. satisfies: A(x) = Integral 1 + A(x)^4 dx.

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A296839 Expansion of e.g.f. tan(x*tan(x/2)) (even powers only).

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A024283 E.g.f. (1/2) * tan(x)^2 (even powers only).

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A117972 Numerator of zeta'(-2n), n >= 0.

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A144696 Triangle of 2-Eulerian numbers.

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A258927 E.g.f. satisfies: A(x) = Integral 1 + A(x)^6 dx.

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A293951 Number of linear extensions of a poset whose Hasse diagram consists of n binary shrubs with type B_n joins.

A302587 a(n) = n! * [x^n] exp(nx)tanh(x).